Cosmetic crossings and Seifert matrices

Cosmetic crossings and Seifert matrices

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.

我们研究亏格为1的纽结中的 cosmetic 交叉（注：cosmetic crossing可能是一种特定的交叉类型，暂未找到合适的中文术语），并根据由塞弗特矩阵确定的纽结不变量得到此类交叉的阻碍。特别地，我们证明对于亏格为1的纽结，亚历山大多项式以及在该纽结上分支的双覆盖的同调为cosmetic交叉提供了阻碍。作为一个应用，我们证明了非缆式纽结的扭曲怀特黑德双倍的平凡交叉猜想。我们还对几类椒盐卷饼纽结以及所有交叉数至多为12的亏格为1的纽结验证了该猜想。